Hyperbolic-space Coulomb field ($E(\xi )=Q/[4\pi {ϵ}_0{R}^2{\sinh }^2(\xi /R)]$)

In static time times round three-dimensional hyperbolic space of curvature radius $R$, a geodesic sphere has area $4\pi R^2{\sinh }^2(\xi /R)$. The Gauss law therefore gives the displayed radial field. It approaches the flat-space inverse-square field near the charge and falls exponentially at large geodesic distance. The curvature changes the area of a sphere, not the conserved flux.